\beginsection{2.3}

Show that $(2+\sqrt2)^{1/2}$ does not represent a rational number.

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$(2+\sqrt2)^{1/2}$ is a solution to the equation
$x^2-2=\sqrt2$.
This can be rewritten as
$$(x^2-2)^2=2$$
which expands to
$$x^4-4x^2+4=2$$
Hence $(2+\sqrt2)^{1/2}$ is a solution to the polynomial
$$x^4-4x^2+2=0$$
By the Rational Zeroes Theorem the only possible rational solutions
are $\pm1$ and $\pm2$, neither of which is an actual solution.
Therefore $(2+\sqrt2)^{1/2}$ is not a rational number.

